Ren-Cang Li, `Relative perturbation theory: (I) Eigenvalue and singular value variations'. SIAM J. Matrix Anal. Appl., vol. 19 (1998), pp. 956{ 982. Home/Search Document Details and Download
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Orthogonal Eigenvectors and Relative Gaps - Dhillon, Parlett (1999) (4 citations) (Correct)
....N r D r N t r may be represented by E 1 N r EFD r FEN t r E 1 (36) 21 5.2 Outer Perturbations We say that E 1 is an outer perturbation and EF is an inner perturbation to N r D r N t r . Outer (multiplicative) perturbations have received considerable attention recently, see [9] 10] [18], 19] and [20] but the e ect of inner perturbations is harder to assess. We reproduce here the simplest results on outer perturbations. Note that E 1 TE 1 I = E 1 (T I)E 1 (E 2 I) If is a nonzero eigenvalue of T then the rst matrix on the right is singular. By Weyl s ....
Ren-Cang Li, `Relative perturbation theory: (I) Eigenvalue and singular value variations'. SIAM J. Matrix Anal. Appl., vol. 19 (1998), pp. 956-982.
Orthogonal Eigenvectors and Relative Gaps - Dhillon, Parlett (2000) (4 citations) (Correct)
....r may be represented by E Gamma1 N r EFD r FEN t r E Gamma1 (36) 5.2 Outer Perturbations We say that E Gamma1 is an outer perturbation and EF is an inner perturbation to N r D r N t r . Outer (multiplicative) perturbations have received considerable attention recently, see [9] 10] [18], 19] and [20] but the effect of inner perturbations is harder to assess. We reproduce here the simplest results on outer perturbations. Note that E Gamma1 TE Gamma1 Gamma I = E Gamma1 (T Gamma I)E Gamma1 (E Gamma2 Gamma I) If is a nonzero eigenvalue of T then the first ....
Ren-Cang Li, `Relative perturbation theory: (I) Eigenvalue and singular value variations '. SIAM J. Matrix Anal. Appl., vol. 19 (1998), pp. 956--982.
Relatively Robust Representations of Symmetric Tridiagonals - Parlett, Dhillon (1999) (4 citations) (Correct)
....L D 1 LD 2 for appropriately chosen diagonal scaling matrices D j = I j ; k j k : See [5] for details. Keeping xed the tridiagonal matrix changes from L L t to D 1 LD 2 D 2 L t D 1 . Outer perturbations corresponding to D 1 have been studied by several authors, see [6] 7] [10], 11] 12] and [19] and cause small relative changes in each eigenvalue. A preliminary study of inner perturbations, corresponding to D 2 , was made in [3] and in his thesis Dhillon 9 introduced a single condition number for inner perturbations, see [4] He applies standard rst order ....
....gives the appropriate expression for relcond(s j ) relcond(s j ; L L t ) It is a somewhat complicated function of the relconds for all the eigenvalues as well as the (relative) separation of the eigenvalues. In order to improve appearances we introduce a little used measure (called in [10]) of relative separation, relsep(a; b) ja bj p jaj jbj (61) and observe that this funtion can reach 1. By this device relcond(s j ) q relcond( j ) X i6=j relcond( i ) relsep 2 ( j ; i ) 1=2 : 62) We conclude with some implications of our de nition of relcond(s ....
Ren-Cang Li, `Relative perturbation theory: (I) Eigenvalue and singular value variations'. SIAM J. Matrix Anal. Appl. vol. 19 (1998), pp. 956{ 982.
An Implementation of the dqds Algorithm (Positive Case) - Parlett, Marques (Correct)
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Ren-Cang Li, `Relative perturbation theory: (I) Eigenvalue and singular value variations'. SIAM J. Matrix Anal. Appl., vol. 19 (1998), pp. 956{ 982.
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